Could Regressing a Stationary Series on a Non-Stationary Series Obtain Meaningful Outcomes?

We have read many papers in the literature and found that some papers report results of regressing a stationary time series on a non-stationary time series (we call it the [Formula: see text] model). However, very few studies, if there are any, examine the [Formula: see text] model and the robustness of inference in such settings remains an open question. To bridge the gap in the literature, in this paper, we investigate whether regressing a stationary time series, [Formula: see text], on a non-stationary time series, [Formula: see text] (that is, [Formula: see text]) could get any meaningful result. To do so, we first conduct a simulation and find regressing a stationary time series on a non-stationary time series could be spurious. Thereafter, we develop the estimation and testing theory for the [Formula: see text] model and find that the statistics [Formula: see text] for testing [Formula: see text] versus [Formula: see text] from the traditional regression model (we call it [Formula: see text] model) does not have any asymptote distribution with [Formula: see text] and [Formula: see text] as [Formula: see text], and thus, it cannot be used for the [Formula: see text] model. We have found other interesting results as shown in our paper. Thus, our paper extends the spurious regression literature to cover a previously unexplored case, thereby contributing to a more comprehensive understanding of time series modeling and inference.